Exercise 2. (a) Let z = a + bi, and express a cos(wt) + b sin(wt) in terms of z, z, elt, and e-it. (Hint: Express cos(wt) and sin(wt) in terms of elt and e-it) (b) In (a), you should have obtained two terms. Show the two terms are complex conjugates of each other. (c) In the chapter on complex numbers, we showed that for any complex number w the real part of w (written as Re[w]) can be expressed as: Re[w] = Use this fact to show that w+w 2 a cos (wt) + b sin(wt) = Re[z. et]. As a consequence of Exercise 2, we can treat real signals that are combinations of cosine and sine waves
Exercise 2. (a) Let z = a + bi, and express a cos(wt) + b sin(wt) in terms of z, z, elt, and e-it. (Hint: Express cos(wt) and sin(wt) in terms of elt and e-it) (b) In (a), you should have obtained two terms. Show the two terms are complex conjugates of each other. (c) In the chapter on complex numbers, we showed that for any complex number w the real part of w (written as Re[w]) can be expressed as: Re[w] = Use this fact to show that w+w 2 a cos (wt) + b sin(wt) = Re[z. et]. As a consequence of Exercise 2, we can treat real signals that are combinations of cosine and sine waves
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please do Exercise 2 part A-C and please show step by step and explain.
![**Exercise 2:**
(a) Let \( z = a + bi \), and express \( a \cos(\omega t) + b \sin(\omega t) \) in terms of \( z, \bar{z}, e^{i \omega t}, \) and \( e^{-i \omega t} \). (Hint: Express \( \cos(\omega t) \) and \( \sin(\omega t) \) in terms of \( e^{i \omega t} \) and \( e^{-i \omega t} \).
(b) In (a), you should have obtained two terms. Show the two terms are complex conjugates of each other.
(c) In the chapter on complex numbers, we showed that for any complex number \( w \) the real part of \( w \) (written as Re[w]) can be expressed as:
\[
\text{Re}[w] = \frac{w + \bar{w}}{2}
\]
Use this fact to show that:
\[
a \cos(\omega t) + b \sin(\omega t) = \text{Re}[z \cdot e^{i \omega t}]
\]
As a consequence of Exercise 2, we can treat real signals that are combinations of cosine and sine waves as the real part of complex signals composed of complex exponentials. It turns out that this enables us to bring in the theory of complex numbers (in particular, complex roots of unity) to gain extremely useful insights into the nature of these signals.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F892e817a-9b32-4eeb-b8fc-5dd7ffde6479%2Fa5d24458-1400-4d40-a4e8-3111a1516d7c%2Fpsh18dg_processed.png&w=3840&q=75)
Transcribed Image Text:**Exercise 2:**
(a) Let \( z = a + bi \), and express \( a \cos(\omega t) + b \sin(\omega t) \) in terms of \( z, \bar{z}, e^{i \omega t}, \) and \( e^{-i \omega t} \). (Hint: Express \( \cos(\omega t) \) and \( \sin(\omega t) \) in terms of \( e^{i \omega t} \) and \( e^{-i \omega t} \).
(b) In (a), you should have obtained two terms. Show the two terms are complex conjugates of each other.
(c) In the chapter on complex numbers, we showed that for any complex number \( w \) the real part of \( w \) (written as Re[w]) can be expressed as:
\[
\text{Re}[w] = \frac{w + \bar{w}}{2}
\]
Use this fact to show that:
\[
a \cos(\omega t) + b \sin(\omega t) = \text{Re}[z \cdot e^{i \omega t}]
\]
As a consequence of Exercise 2, we can treat real signals that are combinations of cosine and sine waves as the real part of complex signals composed of complex exponentials. It turns out that this enables us to bring in the theory of complex numbers (in particular, complex roots of unity) to gain extremely useful insights into the nature of these signals.
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Step by step solution of (a) : Find a,b,cos(wt),sin(wt)
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